Harvey,
Thanx for the answers there, a 1976 Scientific American article
shows that it can be done, but I have not yet read this article.
Here is the actual order 5 perfect cube, which can be magnified
http://cboyer.club.fr/multimagie/English/Perfectcubes.htm
This cube contains all the integers from 1 to 5^3 = 125. There are
109 ways to get the magic sum 315, with its 25 rows, 25 columns,
25 pillars, 4 triagonals, and 30 diagonals. The center number is
63, which is to be expected since the magic sum divided by the
number of planes (5)= 63.
It will be impossible to construct a perfect cube smaller than
order 5.
None of the four existing standard magic cubes of order 3 is
perfect, and the American mathematician Richard Schroeppel in the
Artificial Intelligence Memo of the MIT (1972) proved that a magic
cube of order 4 cannot be perfect.
Perhaps, therefore (returning to fiction), DNA should have let DT
ponder a bit longer to arrive at 63 instead of 42 ;-)
It wasn't easy, for DEEP you know. As any software engineer can
tell you, 42 in base ten is equal to '101010' in base two. This
alternating pattern of ones and zeros could be said to have
illustrated the enormity of DEEP Thought's indecision, starting
from day "Count Zero" to now... about... you know... the Ultimate
Question. After all, the base two representation of 63 is much
more self-assured - being 111111.
Of course, we can get really off-base... as in the original
Hitchhiker's Guide radio show, where the "cave man" put out
Scrabble stones and the sentence "What do you get if you multiply
six by nine?" emerges, and then Arthur says "Six by nine?
Forty-two? You know, I've always felt that there was something
fundamentally wrong with the Universe." -- it is at this point
that a faint and distant voice says "base thirteen!"
42 (base 13) is equal to 54 (base 10). BTW the Sum of the Proper
Divisors of 42
also =54.
Of Course -- Douglas Adams has been quoted as saying " You just
don't write jokes in base 13!" but is there any other connection
between 42 and 63?
Hmm... the divisors of the Positive Integer 126 are 1, 2, 3, 6, 7,
9, 14, 18, 21, 42, 63, 126. That is tantalizing. But the trail has
gone dead after that. 21 and 42 are both power numbers, so it is
no surprise that 63 is way cool.... even if not necessarily the
penultimate.
BTW, to dispel any myths about 42, Douglas Adams also admitted at
times - "It was a joke. It had to be a number, an ordinary,
smallish number, and I chose that one." Binary representations,
base thirteen, Tibetan monks are all complete nonsense. I sat at
my desk, stared into the garden and thought '42 will do' I typed
it out. End of story
